On Tuesday, March 3, 2020 at 5:23:48 AM UTC-7, Bruno Marchal wrote:
>
>
> On 2 Mar 2020, at 14:19, Alan Grayson <[email protected] <javascript:>>
> wrote:
>
>
> Here is where I think you've tried to answer my gravity problem posed on
> another thread. You say there is an infinity of calculations, but what is
> doing the calculations? And why among those multitudes is one set chosen,
> namely our "illusion"? AG
>
>
>
> OK. I have begun the explanation. But here you are again already at the
> step 7 (if you take a look at my paper sane04 general public
> presentation(*))
>
> What is doing the computations? When you implement the computation in the
> physical reality, the computations are done by the relevant digital
> information encoded into physical relations, between physical objects.
> What is doing the computation, when there is no physical universe, is any
> relation in a model of a Turing complete theory. Elementary arithmetic is
> Turing complete, so if you are OK with 2+2=4, or with statement like “there
> is no bigger prime number”, the computation are implemented naturally by
> the representation of those computation in term of true natural number
> relation. It is a bit like a bloc-universe. Time will be accounted by the
> notion of “number of steps of a computation”.
>
> Example. The reality of, say, the computer science statement that" the
> register (a, b, c) contains a", is realised by the physical fact that
> something encoding “a" is put “physically” in a series of physical memories
> (flip-flop, or magnetic disk, etc.).
>
> That same reality is implemented, all by itself, in the arithmetical
> statement that the number
>
> 2^(“a”) * 3^(“b”) * 5^(“c”)
>
> admits only one decomposition into product of power of primes (by the so
> called fundamental theorem of arithmetic, by Euclid), and saying that “a”
> (the representation of ‘a' by some number, which I note “a”) belongs to its
> first place is arithmetically equivalent of saying that 2^”a” divides
> 2^(“a”) * 3^(“b”) * 5^(“c”) , and that 2^”a” + 1 does not divide it.
>
> Of course, that “arithmetise” only a few bit of the computation. Gödel
> missed the Church-Turing thesis, and so was unaware of arithmetising
> “computer science”, but he will got the point later. Meanwhile, he was the
> first one to arithmetise the provability predicate, and later, he will
> understand that his provability predicate is Turing equivalent (with
> respect to computability,; not with respect to provability).
>
> The best might be to download Gödel 1931, for example here:
>
> https://www.jamesrmeyer.com/pdfs/godel-original-english.pdf
>
> The complete arithmetization is done in 40 steps. See below(**). The step
> 44 gives Gödel famous beweisbar predicate Bw(x), which is the one I wrote
> []x, and which is the subject of what I called “theology of machine”,
> mainly the mathematical theory of provability , proved complete by Solovay
> in 1976 (at the modal propositional level). The step 1 is the arithmetical
> definition of “x divides y”, OK?
>
> It has to be long and tedious, as defining provability (and thus
> computability) in arithmetic is like programming a high level language in a
> low level language.
>
> To really swallow this, it will also be necessary to understand well the
> difference between proof and truth. The computations are implemented in the
> truth of arithmetic (in the “model”, or in all models, in the logician
> sense of model (whicht will be brought by Lowenheim and Tarski, Gödel did
> without this, but refer to the intuition, which is simpler, but for our
> concern, we have to take this into account at some point)).
>
> Bruno
>
>
>
> (*) B. Marchal. The Origin of Physical Laws and Sensations. In 4th
> International System Administration and Network Engineering Conference,
> SANE 2004, Amsterdam, 2004.
> http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html
>
>
> (**) Here are the 40 steps, for future reference. Tell me if you can read
> and understand the line 1.
>
Yes. AG
> You might print it, and we can do them step by step too. It *looks*
> difficult, but it is not that much difficult, if you develop a bit of
> familiarisation with the notation (which are not quite standard):
>
> 1. x/y≡(∃z)[z≤x&x=y·z] x is divisible by y.
> 2. Prim(x)≡~(∃z)[z≤x&z≠1&z≠x&x/z]&x>1 x is a prime number.
> 3. 0Prx≡0
> (n+1) Pr x ≡ εy [y ≤ x & Prim(y) & x/y & y > n Pr x]
> n Pr x is the nt h (in order of magnitude) prime number contained in x.34a
> 4. 0!≡1
> (n+1)! ≡ (n+1).n!
> 5. Pr(0) ≡ 0
> Pr(n+1) ≡ εy [y ≤ {Pr(n)}! + 1 & Prim(y) & y > Pr(n)]
> Pr(n) is the nt h prime number (in order of magnitude).
> 6. nGlx≡εy[y≤x&x/(nPrx)y &~x/(nPrx)y+1]
> n Gl x is the nt h term of the series of numbers assigned to the number
> x (for n > 0 and n not greater than the length of this series).
> 7. l(x)≡εy[y≤x&yPrx>0&(y+1)Prx=0]
> l(x) is the length of the series of numbers assigned to x.
> 8. x*y≡εz[z≤[Pr{l(x)+l(y)}]x+y &(n)[n≤l(x)⇒nGlz=nGlx] & (n)[0 < n ≤
> l(y) ⇒ {n+l(x)} Gl z = n Gl y]]
> x * y corresponds to the operation of "joining together" two finite series
> of numbers.
> 9. R(x) ≡ 2x
> R(x) corresponds to the number-series consisting only of the number x
> (for x > 0).
> 10. E(x)≡R(11)*x*R(13)
> E(x) corresponds to the operation of "bracketing" [ 11 and 13 are assigned
> to the basic signs "(" and ")"].
> 11. nVarx≡(∃z)[13<z≤x&Prim(z)&x=z n]&n≠0 x is a variable of nt h type.
> 12. Var(x)≡(∃n)[n≤x&nVarx] x is a variable.
> 13. Neg(x) ≡ R(5) * E(x)
> Neg(x) is the negation of x.
>
> 14. xDisy≡E(x)*R(7)*E(y)
> x Dis y is the disjunction of x and y.
> 15. xGeny≡R(x)*R(9)*E(y)
> x Gen y is the of y by means of the variable x
> (assuming x is a ).
> 16. 0Nx≡x
> (n+1) N x ≡ R(3) * n N x
> n N x corresponds to the operation: " n-fold prefixing of the sign ‘ f’
> before x."
> 17. Z(n) ≡ n N [R(1)]
> Z(n) is the number-sign for the number n.
> 18. Typ1′(x)≡(∃m,n){m,n≤x&[m=1∨1Varm] &x=nN[R(m)]} 34b
> x is a sign of first type.
> 19. Typn(x)≡[n=1&Typ1′(x)]∨[n>1
> & (∃v){v ≤ x & n Var v & x = R(v)}] x is a sign of nth type.
> 20. Elf(x) ≡ (∃y,z,n)[y,z,n ≤ x & Typn(y) & Typn+1(z) & x = z * E(y)] x is
> an elementary formula.
> 21. Op(x,y,z)≡x=Neg(y)∨x=yDisz ∨ (∃v)[v ≤ x & Var(v) & x = v Gen y]
> generalization
> variable
>
> 22. FR(x)≡(n){0<n≤l(x)⇒Elf(nGlx) ∨(∃p,q)[0<p,q<n&Op(nGlx,pGlx,qGlx)]}
> &l(x)>0
> x is a series of formulae of which each is either an
> or arises from those preceding by the operations of , disjunction
> and generalization.
> 23. Form(x) ≡ (∃n){n ≤ (Pr[l(x)2])x · [l(x)]2 & FR(n) & x = [l(n)] Gl n}
> 35 x is a formula (i.e. last term of a series of formulae n).
> 24. v Geb n,x ≡ Var(v) & Form(x)
> & (∃a,b,c)[a,b,c ≤ x & x = a * (v Gen b) * c & Form(b) & l(a)+1 ≤ n
> ≤ l(a)+l(v Gen b)]
> The variable v is bound at the nt h place in x.
> 25. vFrn,x≡Var(v)&Form(x) &v=nGlx&n≤l(x)&~(vGebn,x)
> The variable v is free at the nt h place in x.
> 26. vFrx≡(∃n)[n≤l(x)&vFrn,x]
> v occurs in x as a free variable.
> 27. Sux( n⁄y )≡εz{z≤[Pr(l(x)+l(y))]x+y
> &[(∃u,v)u,v≤x&x=u*R(bGlx)*v &z=u*y*v&n=l(u)+1]} Su x( n ⁄y ) derives from
> x on substituting y in place of the nt h term of x
> (it being assumed that 0 < n ≤ l(x)).
> 28. 0Stv,x≡εn{n≤l(x)&vFrn,x &~(∃p)[n<p≤l(x)&vFrp,x]} (k+1) St v,x ≡ εn
> {n < k St v,x & v Fr n,x
> & (∃p)[n < p < k St v,x & v Fr p,x]}
> k St v,x is the (k+1) t h place in x (numbering from the end of formula
> x) at which v is free in x (and 0, if there is no such place.)
> elementary formula
> negation
>
> 29. A(v,x)≡εn{n≤l(x)&nStv=0}
> A(v,x) is the number of places at which v is free in x.
> 30. Sb0(x v⁄y )≡x
> Sbk+1(x ⁄y )≡Su{Sbk(x ⁄y )}( ⁄y )
> v v k St v, x 31. Sb(x v⁄y )≡SbA(v,x)(x v⁄y )36
> Sb(x v ⁄y ) is the concept Subst a( v ⁄b ), defined above.3 7
> 32. xImpy≡[Neg(x)]Disy
> x Con y ≡ Neg{[Neg(x)] Dis [Neg(y)]} x Aeq y ≡ (x Imp y) Con (y Imp x)
> v Ex y ≡ Neg{v Gen [Neg(y)]}
> 33. nThx≡εn{y≤x(xn) & (k)≤l(x)⇒(kGlx≤13&kGly=kGlx)∨ (kGlx>13&kGly=kGlx
> ·[1Pr(kGlx)]n)]}
> nThxisthenth type-liftofx(inthecasewhenxandnThxare formulae).
> To the axioms I, 1 to 3, there correspond three determinate numbers, which
> we denote by z1, z2, z3, and we define:
> 34. Z–Ax(x)≡(x=z1 ∨x=z2 ∨x=z3)
> 35. A1-Ax(x)≡(∃y)[y≤x&Form(y)&x=(yDisy)Impy]
> x is a formula derived by substitution in the axiom-schema II, 1.
> Similarly A2-Ax, A3-Ax, A4-Ax are defined in accordance with the axioms II,
> 2 to 4.
> 36. A-Ax(x) ≡ A1-Ax(x) ∨ A2-Ax(x) ∨ A3-Ax(x) ∨ A4-Ax(x)
> x is a formula derived by substitution in an axiom of the
> sentential calculus.
>
> 37. Q(z,y,v) ≡ ~(∃n,m,w)[n ≤ l(y) & m ≤ l(z)
> & w ≤ z & w = m Gl z & w Geb n,y & v Fr n,y]
> z contains no variable bound in y at a position where v is free.
> 38. L1-Ax(x) ≡ (∃v,y,z,n){v,y,z,n ≤ x & n Var v & Typ vn(z)
> &Form(y)&Q(z,y,v)&x=(vGeny)Imp[Sb(y ⁄z )]}
> x is a formula derived from the axiom-schema III, 1 by substitution.
> 39. L2-Ax(x) ≡ (∃v,q,p){v,q,p ≤ x & Var(v) & Form(p) & v Fr p & Form(q)
> & x = [v Gen (p Dis q)] Imp [p Dis (v Gen q)]}
> x is a formula derived from the axiom-schema III, 2 by substitution.
> ...
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